Are these two definitions of separable extension equivalent?

Question

Definition1 - wikipedia

Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable.

Definition 2 - Lang

Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff $[E:F]=[E:F]_{sep}$.

I know if the extension is finite, then these two definitions are equivalent. However, if the extension is not finite, are these definitions still equivalent? And which is the standard one?

Answer 1

The statements agree when the extension is finite as you point out. However, the notion of degree is not meaningful in the context of infinite extensions.

Thus, the second characterization does not exist for infinite extensions.